UC Irvine UC Irvine Electronic Theses and Dissertations Title New Applications of the Nearest-Neighbor Chain Algorithm Permalink https://escholarship.org/uc/item/6nt3g7d4 Author Mamano Grande, Nil Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE New Applications of the Nearest-Neighbor Chain Algorithm DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Computer Science by Nil Mamano Grande Dissertation Committee: Professor Michael T. Goodrich, Chair Professor David Eppstein Professor Sandy Irani 2019 c 2019 Nil Mamano Grande TABLE OF CONTENTS Page LIST OF FIGURES v LIST OF TABLES vii LIST OF ALGORITHMS viii ACKNOWLEDGMENTS ix CURRICULUM VITAE x ABSTRACT OF THE DISSERTATION xii 1 Introduction 1 1.1 Greedy algorithms . 3 1.2 Local greedy . 4 1.3 Global–local equivalence . 6 1.4 Background . 8 1.4.1 Hierarchical clustering . 8 1.4.2 Nearest-neighbor chain algorithm . 10 2 Global–Local Equivalence 13 2.1 Combinatorial optimization problems . 16 2.1.1 Maximum-weight independent set . 22 2.2 Multi-fragment TSP . 25 2.3 Shortest common superstring . 28 2.4 Conclusions . 36 3 Nearest-Neighbor Chain Algorithms 37 3.1 Preliminaries: nearest-neighbor data structures . 37 3.2 Soft nearest-neighbor data structure . 40 3.2.1 Implementation . 41 3.2.2 Choice of parameters . 43 3.2.3 The m-way soft nearest-neighbor data structure . 47 3.2.4 The closest pair problem . 47 3.3 Geometric TSP . 48 ii 3.3.1 Related work . 49 3.3.2 Soft nearest-neighbor chain algorithm . 50 3.3.3 Nearest-neighbor chain algorithm for general graphs . 58 3.4 Steiner TSP . 58 3.5 Motorcycle graphs . 60 3.5.1 Algorithm description . 62 3.5.2 Analysis . 64 3.5.3 Special cases and remarks . 66 3.6 Server cover . 67 3.6.1 Related work . 67 3.6.2 Global–local equivalence . 68 3.6.3 Algorithm description . 69 3.6.4 Analysis . 72 3.6.5 Global greedy in higher dimensions . 76 3.7 Shortest common superstring . 78 3.7.1 Background . 78 3.7.2 First-choice chain algorithm . 79 3.7.3 Analysis . 80 3.7.4 Discussion . 82 3.8 Geometric matching . 83 3.8.1 Soft nearest-neighbor chain algorithm . 84 3.9 Combinatorial optimization problems . 85 3.9.1 Problems . 88 3.10 Conclusions . 91 4 Symmetric Stable Matching 93 4.1 Background . 93 4.2 Symmetric stable matching . 95 4.2.1 Greedy algorithm . 96 4.2.2 Local greedy algorithm . 97 4.2.3 First-choice chain algorithm . 99 4.3 Geometric model . 100 4.4 Geographic model . 101 4.5 Narcissistic k-attribute model . 103 4.5.1 The 2-attribute case . 105 4.6 The stable roommates problem . 107 4.7 Conclusions . 108 5 Proximity Data Structures In Graphs 111 5.1 Introduction . 111 5.1.1 Background . 112 5.1.2 Our contributions . 115 5.1.3 Applications . 116 5.2 Nearest-neighbor data structure . 117 5.2.1 Preprocessing . 118 iii 5.2.2 Queries . 119 5.2.3 Updates . 121 5.3 Extensions and related data structures . 122 5.4 Experiments . 123 5.4.1 Implementation details . 123 5.4.2 Results . 124 5.5 Conclusions . 126 6 Stable-Matching Voronoi Diagrams 128 6.1 Introduction . 128 6.2 The geometry of stable-matching Voronoi diagrams . 133 6.3 Combinatorial complexity . 137 6.3.1 Upper bound on the number of faces . 137 6.3.2 Upper bound on the number of edges and vertices . 138 6.3.3 Lower bound . 140 6.4 Algorithms . 146 6.4.1 Discrete algorithm . 148 6.4.2 Polygonal convex distance functions . 159 6.4.3 Discretized plane . 165 6.5 Conclusions . 165 7 Stable Redistricting 167 7.1 Introduction . 167 7.2 Geographic setting . 169 7.2.1 Circle-growing algorithm . 171 7.2.2 Experiments . 172 7.3 Grid Setting . 175 7.3.1 Circle-growing algorithm . 177 7.3.2 Distance-sorting algorithms . 181 7.3.3 Experiments . 184 7.4 Center location . 190 7.4.1 Stable k-means with weighted centroids . 191 7.5 Conclusions . 193 Bibliography 197 iv LIST OF FIGURES Page 1.1 Global–local equivalence in maximum-weight matching. 7 1.2 Lack of global–local equivalence in hierarchical clustering with centroid distance. 10 2.1 Proof of global–local equivalence in maximum-weight matching. 15 2.2 Proof via hybrid method. 21 2.3 Interaction graph for a set cover instance. 22 2.4 Independent set instance where global greedy and local greedy differ. 23 2.5 Independent set instance where local greedy and soft local greedy differ. 24 2.6 Proof of Lemma 2.17. 32 2 3.1 Soft nearest-neighbor parameters in (R ;L2). .................... 46 3.2 Illustration of the SNNC algorithm for geometric TSP. 54 3.3 Illustration of the NNC algorithm for motorcycle graphs. 62 3.4 Global–local equivalence counterexample for server cover. 69 3.5 Illustration of the NNC algorithm for server cover. 72 3.6 Setting in the proof of Lemma 3.27. 75 3.7 Tight example for the 2-approximation of NNC for server cover. 76 3.8 Bad instance for the greedy algorithm for server cover in 2D. ..